Projector-based renormalization approach to electron-hole-photon systems in their nonequilibrium steady state

We present an extended version of the projector-based renormalization method that can be used to address not only equilibrium but also nonequilibrium situations in coupled fermion-boson systems. The theory is applied to interacting electrons, holes, and photons in a semiconductor microcavity, where the loss of cavity photons into vacuum is of particular importance. The method incorporates correlation and fluctuation processes beyond mean-field theory in a wide parameter range of detuning, Coulomb interaction, light-matter coupling, and damping, even in the case when the number of quasiparticle excitations is large. This enables the description of exciton and polariton formation and their possible condensation through spontaneous phase symmetry breaking by analyzing the ground-state, steady-state, and spectral properties of a rather generic electron-hole-photon Hamiltonian, which also includes the coupling to two fermionic baths and a free-space photon reservoir. Thereby, the steady-state behavior of the system is obtained by evaluating expectation values in the long-time limit by means of the Mori-Zwanzig projection technique. Tracking and tracing different order parameters, the fully renormalized single-particle spectra and the steady-state luminescence, we demonstrate the Bose-Einstein condensation of excitons and polaritons and its smooth transition when the excitation density is increased.

Figure 1

Parameter $\mu$ characterizing the steady state description (13) and (14), where $\mu$ becomes the chemical potential in equilibrium. Pictured here is $\mu$ as a function of ${\mu }_B$ for different values of $\gamma$ at fixed $\kappa ={10}^{-5}$ (left panels) and, likewise, for different values of $\kappa$ at fixed $\gamma =0.1$ (right panels). The two upper panels refer to detuning $d=3.5$, the lower panels refer to $d=-0.5$; note the different scales of the ordinates.

Figure 2

Steady-state parameter $\mu$ as a function of the total density of excitations in the e-h-p microcavity system, $n_{\mathrm{exc}}$ from (180). Shown are results for $d=3.5$ (top) and $d=-0.5$ (bottom) at a fixed value of $\kappa$ (left) and $\gamma$ (right).

Figure 3

Excitonic order parameter ${\mathrm{\Delta }}_X$ [see Eq. (243)], reflecting the exciton contribution to the condensate in the steady state. Depicted are the real parts (solid lines) and imaginary parts (dashed lines) of ${\mathrm{\Delta }}_X$ as a function of $n_{\mathrm{exc}}$ for large (top) and small (bottom) detuning at fixed $\kappa$ (left) and $\gamma$ (right).

Figure 4

Photonic order parameter ${\mathrm{\Delta }}_{ph}$ [see Eq. (243)], reflecting the photon contribution to the condensate in the steady state. Here, ${\mathrm{\Delta }}_{ph}$ is given as a function of $n_{\mathrm{exc}}$ for $d=3.5$ (top) and $d=-0.5$ (bottom) at fixed $\kappa ={10}^{-5}$ (left) and $\gamma =0.1$ (right).

Figure 5

Electron-hole pairing amplitude $d_{\mathbf{k}}$ [see Eq. (33)], indicating exciton condensation. Shown is an intensity plot of its real part in the momentum-density plane for semimetallic ($d=3.5$; left panels) and semiconducting ($d=-0.5$; right panels) situations.

Figure 6

Exciton pairing amplitude $d_{\mathbf{k}}$. Shown is the intensity plot of its imaginary part in the momentum-density plane for $d=3.5$ (left) and $d=-0.5$ (right).

Figure 7

Intensity of the photon field $\langle {\psi }_{\mathbf{q}}^{†}{\psi }_{\mathbf{q}}\rangle$ in the momentum-density plane for detunings $d=3.5$ (left) and $d=-0.5$ (right).

Figure 8

Electron single-particle spectrum in the steady state of the considered e-h-p microcavity system. The quasiparticle band dispersion clearly appears in the intensity plot of the coherent part of the fully renormalized spectral function, $A_{e,\mathrm{coh}}(\mathbf{k},\omega )$ given by Eq. (211). Results are given for typical semimetallic ($d=3.5$, left) and semiconducting ($d=-0.5$, right) situations. Here, the excitation density $n_{\mathrm{exc}}=0.8$. Note that the frequency is measured from $\mu$.

Figure 9

Steady-state luminescence of the e-h-p microcavity system under consideration. Shown is an intensity plot of its incoherent part, $S_{\mathrm{inc}}(\mathbf{q},\omega )$ given by Eq. (234), at $n_{\mathrm{exc}}=0.8$, for $d=3.5$ (left) and $d=-0.5$ (right).